Coverings by convex bodies and inscribed balls

Abstract

Let $H$ be a Hilbert space. For a closed convex body $A$ denote by $r(A)$ the supremum of radiuses of balls, contained in $A$. We prove, that ${\sum }_{n=1}^{\infty }r(A_n)\ge r(A)$ for every covering of a convex closed body $A\subset H$ by a sequence of convex closed bodies $A_n$, $n\in \mathbb{N}$. It looks like this fact is new even for triangles in a 2-dimensional space.

6 pages